GENERATING NON-NOETHERIAN MODULES CONSTRUCTIVELY
THIERRY COQUAND, HENRI LOMBARDI, CLAUDE QUITTÉ
Abstract. In [6], Heitmann gives a proof of a Basic Element Theorem, which has as
corollaries some versions of the “Splitting-off” theorem of Serre and the Forster-Swan
theorem in a non Noetherian setting. We give elementary and constructive proofs of such
results. We introduce also a new notion of dimension for rings, which is only implicit in
[6] and we present a generalisation of the Forster-Swan theorem, answering a question left
open in [6].
1. Zariski spectrum and Krull dimension
Let R be a commutative ring with unit. Following Joyal [7], we define the Zariski
spectrum of R as the distributive lattice generated by symbols D(f ), f ∈ R and relations
D(0) = 0
D(1) = 1 D(f g) = D(f ) ∧ D(g)
D(f + g) ≤ D(f ) ∨ D(g)
We write D(f1 , . . . , fm ) for D(f1 ) ∨ · · · ∨ D(fm ). For m = 0 we have D() = 0. It can be
shown directly that
D(g1 ) ∧ · · · ∧ D(gn ) ≤ D(f1 , . . . , fm )
holds if and only if the monoid generated by g1 , . . . , gn meets the ideal generated by
f1 , . . . , fm [1]. Thus D(f1 , . . . , fm ) can be defined as the radical of the ideal generated
by f1 , . . . , fm (with inclusion as ordering), and we have a point-free and elementary description of the basic open sets of the Zariski spectrum of R.
In [2] we present the following elementary characterization of Krull dimension. If a ∈ R
we define the boundary of a as being the the ideal Na generated by a and the elements b
such that ab is nilpotent (or equivalently D(ab) = 0).
Theorem 1.1. The dimension of R is <n + 1 if and only if for all a ∈ R the dimension
of R/Na is <n.
This can actually be taken as a constructive definition of Krull dimension, if we define a
ring R to be of dimension <0 if and only if R is trivial. This inductive definition of being of
dimension <n is then equivalent to the usual definition that there is no strictly increasing
chain of prime ideals of length n [2]. In [1] it is shown, in an elementary and constructive
way, that the dimension of a polynomial ring with n variables over a field is ≤ n.
2. The stable range theorem
All the arguments will be based on the following trivial remark, that we state explicitely
since it will motivate the notion of dimension that we present in section 4.
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Lemma 2.1. If b is nilpotent then 1 = D(b1 , . . . , bk ) ∨ D(b) implies 1 = D(b1 , . . . , bk ).
More generally, if b ∈ R is nilpotent in R[a−1 ] then D(a) ≤ D(b1 , . . . , bk ) ∨ D(b) implies
D(a) ≤ D(b1 , . . . , bk ).
Proof. Indeed, b is nilpotent if and only if D(b) = 0. Likewise, b ∈ R is nilpotent in R[a−1 ]
if and only if D(ab) = 0, and D(a) ≤ D(b1 , . . . , bk ) ∨ D(b) implies D(a) ≤ D(b1 , . . . , bk ) ∨
D(ab).
We shall need also the following two remarks.
Lemma 2.2. D(y + b) ∨ D(yb) = D(y) ∨ D(b)
Lemma 2.3. If by is nilpotent then 1 = D(b1 , . . . , bk , b, y) implies 1 = D(b1 , . . . , bk , b + y).
More generally, if by ∈ R is nilpotent in R[a−1 ] then D(a) ≤ D(b1 , . . . , bk , b, y) implies
D(a) ≤ D(b1 , . . . , bk , b + y).
Proof. By Lemmas 2.1 and 2.2.
Notice that the nilpotent hypothesis was used only to invoke lemma 2.1.
Our inductive definition of dimension allows more perspicuous proofs. For instance, as
a motivation of our method, here is a proof of the “Stable Range” theorem.
Theorem 2.4. If the dimension of R is < n and 1 = D(a, b1 , . . . , bn ) there exists x1 , . . . , xn
such that 1 = D(b1 + ax1 , . . . , bn + axn ).
Proof. The proof is by induction on n. This is clear if n = 0, since in this case the ring is
trivial and we have 1 = D(). If n > 0, let I be the ideal boundary of bn . We have bn ∈ I
and the dimension of R/I is <n − 1. By induction, we can find x1 , . . . , xn−1 such that
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 )
in R/I. This means that there exists xn such that D(bn xn ) = 0 and
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 ) ∨ D(bn ) ∨ D(xn )
Since
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 ) ∨ D(bn ) ∨ D(a)
this implies by distributivity
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 ) ∨ D(bn ) ∨ D(axn )
hence the result by Lemma 2.3.
It follows then for instance directly that a stably free module of rank ≥ n over a ring
of dimension <n is free [9], without any noetherian hypotheses. We can in the same way
prove Kronecker’s theorem about algebraic sets [3, 8].
We shall need a variation on this result. If L ∈ Rn is a vector (a1 , . . . , an ) we write D(L)
for D(a1 , . . . , an ).
Lemma 2.5. If a ∈ R and the dimension of R[a−1 ] is <n then for any L ∈ Rn there
exists X ∈ Rn such that D(a) ≤ D(L − aX). Furthermore, we can find X of the form
aY, Y ∈ Rn .
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Proof. We let L be (b1 , . . . , bn ) and we reason by induction on n. This is clear if n = 0.
If n > 0 let N be the ideal boundary of bn in R[a−1 ], and I the ideal N ∩ R. It can
be checked that I that (R/I)[a−1 ] is isomorphic to R[a−1 ]/N . Hence we can apply the
induction hypothesis to R/I and compute (x1 , . . . , xn−1 ) ∈ Rn−1 such that
D(a) ≤ D(b1 − ax1 , . . . , bn−1 − axn−1 )
in R/I. In turn, this means that we can find xn such that D(axn bn ) = 0 and
D(a) ≤ D(b1 − ax1 , . . . , bn−1 − axn−1 ) ∨ D(bn ) ∨ D(xn )
in R. This implies
D(a) ≤ D(b1 − ax1 , . . . , bn−1 − axn−1 ) ∨ D(bn ) ∨ D(axn )
hence the result by Lemma 2.3. Finally, as we can apply the result with a2 instead of a
since R[a−2 ] = R[a−1 ] and D(a) = D(a2 ), we get the result with X = aY .
Corollary 2.6. Let M be a n × n matrix of element in R and δ its determinant. If
the dimension of R[δ −1 ] is <n then for each C ∈ Rn there exists X ∈ Rn such that
D(δ) ≤ D(M X − C). Furthermore we can find X of the form δY, Y ∈ Rn .
f be the adjoint matrix of M , and
Proof. The proof is based on Cramer formulae. Let M
fC. We have then M
f(M X − C) = δX − L for an arbitrary column vector X ∈ Rn .
L=M
Hence the ideal generated by the coordinates of δX − L is included in the one generated
by the coordinates of M X − C, and
D(δX − L) ≤ D(M X − C)
By Lemma 2.5 we can find one X = δY ∈ Rn such that D(δ) ≤ D(δX − L), and hence
D(δ) ≤ D(M X − C) as desired.
3. A Basic Element Theorem
Let F be a rectangular matrix of elementsWin R of columns C0 , C1 , . . . , Cp , and G the
matrix of columns C1 , C2 , . . . , Cp . Let ∆n be ν D(ν) where ν varies over all minors of F
of order n.
Theorem 3.1. Fix 0<n ≤ p. Suppose that for each minor ν of G of order n the ring R[ν −1 ]
is of dimension <n. Then there exist t1 , . . . , tp such that ∆n ≤ D(C0 + t1 C1 + · · · + tp Cp )
and D(C0 ) ≤ D(C0 + t1 C1 + · · · + tp Cp ).
W
Proof. Let ∆0n be ν D(ν) where ν varies over all minors of G of order n. For any t1 , . . . , tp
and any minor ν of F which uses the column C0 we have D(ν) ≤ ∆0n ∨ D(C0 + t1 C1 + · · · +
tp Cp ) and so it suffices to prove the theorem with ∆n replaced by ∆0n .
It is also enough to show that for one minor ν of G of order n we can find t1 , . . . , tp such
that D(ν) ≤ D(C0 + t1 C1 + · · · + tp Cp ) and D(C0 ) ≤ D(C0 + t1 C1 + · · · + tp Cp ) because
we can then apply this successively to all minors of G of order n. But this is a direct
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consequence of Corollary 2.6: we find t1 , . . . , tp (with ti = 0 for the columns outside the
minor ν) that are multiple of ν and such that
D(ν) ≤ D(C0 + t1 C1 + · · · + tp Cp )
Since t1 , . . . , tp are all multiple of ν we have also D(C0 ) ≤ D(ν) ∨ D(C0 + t1 C1 + · · · + tp Cp )
and hence D(C0 ) ≤ D(C0 + t1 C1 + · · · + tp Cp ) as required.
Corollary 3.2. Suppose that 1 = ∆1 and that for each k > 0 and for each minor ν of G
of order n the ring R[ν −1 ]/∆n+1 is of dimension <n. Then there exist t1 , . . . , tp such that
the vector C0 + t1 C1 + · · · + tp Cp is unimodular.
Proof. In the statement of this corollary, we identify ∆l with its corresponding radical
ideal. Using Theorem 3.1, we define a sequence of vectors C0i , i = 1, . . . with C01 = C0 .
For each k > 0, reasoning in R/∆k+1 , we build C0k+1 of the form C0k + u1 C1 + · · · + up Cp
such that ∆k ≤ D(C0k+1 ) and D(C0k ) ≤ D(C0k+1 ) in R/∆k+1 . This means that we have, in
R
D(C0k ) ∨ ∆k ≤ D(C0k+1 ) ∨ ∆k+1
Hence the result since ∆1 = 1 and ∆k = 0 for n large enough.
From this follows directly, as in [4, 6], a version of Serre’s “Splitting-off” theorem and
the Forster-Swan theorem with Krull dimension and without noetherian hypothesis. For
instance, here is a version of Forster’s theorem [5].
Corollary 3.3. Let M be a module which is finitely generated over a ring R of dimension
≤ d. If M is locally generated by r elements then M can be generated by d + r elements.
Proof. M is a quotient of a finitely presented module M 0 which has a Fitting ideal of order
r which contains 1, and we can as well suppose that M 0 = M . Let m0 , m1 , . . . , mp be a
system of generators of M and F be a presentation matrix of M . If p ≥ d + r we have
1 = ∆d+1 (F ) and using theorem 3.1, we can find t1 , . . . , tp such that M is generated by
m1 − t1 m0 , . . . , mp − tp m0 . Hence we can generate M by p elements.
Using Corollary 3.2, one could give a more sophisticated version of this result, as in
[6]. The next section shows how to recover an improved version of these theorems with
Heitmann’s notion of j-spectrum [6], also without noetherian hypothesis.
4. A New Notion of Dimension
By analogy with our inductive definition of Krull dimension, we define now when a ring
R is of H-dimension <n. Let J be the Jacobson radical of R, that is the ideal of elements
x such that 1 − xy is invertible for all y ∈ R. We redefine Na , ideal boundary of a, as the
ideal generated by a and the elements b such that ab ∈ J. Thus we replace the radical of R,
intersection of all prime ideals, by its Jacobson radical, intersection of all maximal ideals.
A ring of H-dimension <0 is a trivial ring, and R is of H-dimension <n + 1 if and only
if for any a ∈ R the H-dimension of R/Na is <n. The proofs of all our previous results
go through directly with this new definition of dimension. This follows from the following
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elementary result, which shows that Lemma 2.1 holds with our new notion, and also that
this new notion is in some sense optimal.
Lemma 4.1. b ∈ J if and only if for all b1 , . . . , bk , we have D(b1 , . . . , bk ) = 1 whenever
D(b1 , . . . , bk ) ∨ D(b) = 1. More generally b ∈ R is in the Jacobson radical of R[a−1 ] if and
only if D(a) ≤ D(b1 , . . . , bk ) whenever D(a) ≤ D(b1 , . . . , bk ) ∨ D(b).
Proof. Assume that D(b1 , b) = 1 implies D(b1 ) = 1. Since 1 − by + by = 1 we get that
1 − by is invertible for all y, that is b ∈ J. Conversely if b ∈ J and y1 b1 + · · · + yk bk + by = 1
then, since 1 − by is invertible, we have 1 = D(b1 , . . . , bk ). The proof of the second part is
similar.
We get in this way a version of the Basic Element Theorem 3.1 with H-dimension instead.
Theorem 4.2. Fix n ≤ p. Suppose that for each minor ν of G of order n the ring R[ν −1 ]
is of H-dimension <n. Then there exist t1 , . . . , tp such that ∆n ≤ D(C0 + t1 C1 + · · · + tp Cp )
and D(C0 ) ≤ D(C0 + t1 C1 + · · · + tp Cp ).
Corollary 4.3. Suppose that for each n ≤ p and for each minor ν of G of order n the
ring R[ν −1 ]/∆n+1 is of H-dimension <n. Then there exist t1 , . . . , tp such that the vector
C0 + t1 C1 + · · · + tp Cp is unimodular.
Our notion of H-dimension is only implicit in [6]. Heitmann introduces instead the jspectrum of R which is the closure of the maximal spectrum of R in the patch topology,
with the topology induced by Zariski topology, and defines the j-dimension of R to be the
(Krull) dimension of the j-Spec(R). Recall that Heitmann’s j-spectrum is not the usual
one given in the literature. This new spectrum was introduced in [6] in order to deal with
the non Noetherian case.
Proposition 4.4. If the j-dimension of R is ≤ n then its H-dimension is also ≤ n.
Proof. By induction on n. Let X be j-Spec R. For a ∈ R, the boundary ideal Na
corresponds to a closed subset Y = V (a) ∩ D(a) ∩ X = Spec(R/Na ) (as in [6], ¯ indicates
closure in the Zariski topology). The subset Y being closed, its closed (=maximal) points
are exactly the points in Max(R) ∩Y . This implies j-Spec(R/Na )⊆ Y ∩ X. It is then
enough to show that the (Krull) dimension of Y ∩ X is <n. But we have Y ∩ X ⊆
V (a) ∩ X ∩ D(a) ∩ X ∩ X which has dimension <n by Lemma 1.2 in [6] (this is the
boundary of D(a) ∩ X in X).
This means that the H-dimension is ≤ the j-dimension. We conjecture that the two
notions of dimension, H-dimension and j-dimension, may differ in general. So our results
could improve those of [6] in some cases. Nevertheless we think that the main importance
of our framework is the constructive and elementary character of our proofs. The next
section illustrates this point, by answering a question left open at the end of [6].
5. A generalisation of the Forster-Swan theorem
The following result is a refinement of Theorem 2.4.
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Lemma 5.1. Assume L, L1 . . . , Ln are vectors of the same length. If H-dim(R)<n and
1 = D(a, b1 , . . . , bn ) ∨ D(L)
then there exist x1 , . . . , xn , all multiple of a, such that
1 = D(b1 + ax1 , . . . , bn + axn ) ∨ D(L + x1 L1 + · · · + xn Ln )
Proof. The proof is by induction on n. This is clear if n = 0. If n > 0, let I be the
ideal boundary of bn . We have bn ∈ I and H-dim(R/I)<n − 1. By induction, we can find
x1 , . . . , xn−1 multiple of a such that
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 ) ∨ D(L + x1 L1 + · · · + xn−1 Ln−1 )
in R/I. This means that there exists y such that bn y ∈ J and
1 = D(b1 + ax1 , . . . , bn−1 + axn−1 ) ∨ D(L + x1 L1 + · · · + xn−1 Ln−1 ) ∨ D(bn ) ∨ D(y)
in R. Take xn = ay. Then xn is a multiple of a and we claim
1 = D(b1 + ax1 , . . . , bn + axn ) ∨ D(L + x1 L1 + · · · + xn Ln )
Indeed, if X is the right hand-side, we have
X ∨ D(a) = D(a, b1 , . . . , bn ) ∨ D(L) = 1
since all x1 , . . . , xn are multiple of a, but also, since xn is a multiple of y
X ∨ D(y) = D(b1 + ax1 , . . . , bn−1 + axn−1 , bn ) ∨ D(L + x1 L1 + · · · + xn−1 Ln−1 ) ∨ D(y)
and hence X ∨ D(y) = 1. Also
X ∨ D(bn ) = X ∨ D(bn ) ∨ D(axn ) = 1
so that X ∨ D(bn y) = 1 and hence X = 1 since we can apply Lemma 4.1 to bn y ∈ J.
We can now prove a variation of Theorem 4.2.
Theorem 5.2. Fix n ≤ p. Suppose that R is of H-dimension <n and ∆n = 1. Then there
exist t1 , . . . , tp such that 1 = D(C0 + t1 C1 + · · · + tp Cp ).
W
Let ∆n (C0 ; G) be ν D(ν) where ν varies over all minors of F of order n using the
column C0 .
Lemma 5.3. If ν is a minor of G of order n using the columns Ci1 , . . . , Cin and
∆n (C0 ; G) ∨ D(ν) = 1
and H-dim(R)< n then there exist x1 , . . . , xn such that
∆n (C0 + x1 Ci1 + · · · + xn Cin ; G) = 1
Proof. This follows from lemma 5.1: we take a to be ν, bk to be the minor obtained from
ν by replacing Cik by C0 , L to be the vector of all remaining minors in ∆n (C0 ; G), and Lk
is obtained by replacing C0 by Cik in L.
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For proving Theorem 5.2, we write
∆n = ∆n (C0 ; G) ∨
_
D(ν)
ν
where ν varies over all minors of G of order n. Applying Lemma 5.3 to suitable quotients
of R, we can eliminate successively these minors, replacing at each step C0 by a vector of
the form C0 + t1 C1 + · · · + tp Cp . At the end, we get t1 , . . . , tp such that ∆n (C0 + t1 C1 +
· · · + tp Cp ; G) = 1 and this implies D(C0 + t1 C1 + · · · + tp Cp ) = 1
We can now state the following result, which can be proved from Theorem 5.2 by arguments similar to the ones for Corollary 3.3.
Theorem 5.4. If H-dim(R)≤ d and if M is a finitely generated module over R which is
locally generated by r elements, then M is generated by d + r elements.
Swan’s theorem [10], which itself generalises Forster’s theorem [5], can be seen as a
special case when the maximal spectrum of R is Noetherian. This improves also on [6]
that assumes instead j-dim(R[a−1 ])≤ d for all a ∈ R, and on [11] that obtained r(d0 + 1)
generators with d0 = j-dim(R).
Acknowledgement
We thank the referee for his careful reading of this paper.
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